52 Cards and 52 Factorial: Why Every Shuffle Is Almost Unique

Pick up a deck of cards and shuffle it properly. The order you produce has, in all likelihood, never existed before in the history of card playing. That claim sounds like an exaggeration, but it is a straightforward consequence of how large the numbers become when you count arrangements. This guide explains the structure of a standard deck and the mathematics of its orderings, along with what it means for shuffling. It is intended for adults and is purely informational.

How a standard deck is built

The deck used in most English-speaking countries has 52 cards. They are divided into four suits, which are spades, hearts, diamonds and clubs, and each suit contains 13 ranks: the numbers 2 to 10, then jack, queen, king and ace. Many sets also include one or two jokers, which are usually removed for games such as poker and bridge. The suit symbols and court cards come from French card designs, which became dominant in much of the world. For the long story of how the deck developed, see A Short History of Playing Cards.

Counting the orderings

How many ways can 52 different cards be arranged in a row? Think of it one position at a time:

1. The first position can hold any of the 52 cards.
2. The second can hold any of the remaining 51.
3. The third has 50 choices, and so on, down to the last position, which has just one card left.

Multiply all these together: 52 × 51 × 50 × ... × 2 × 1. This is written 52! and read as 52 factorial. It is roughly 8.07 followed by 67 zeros, or about 8.07 × 10^67. To put that into perspective, the number of seconds that have passed since the universe began is vastly smaller, being on the order of 10^17. So even if every person on Earth shuffled a deck every second for the whole of history, the number of orderings produced would be a tiny fraction of those possible.

What this means for a shuffle

A well-shuffled deck is, in practice, in an order nobody has seen before. That is why matching two shuffled decks card for card is, for all practical purposes, impossible. It also explains why it is so hard to predict a card from a properly shuffled deck.

How much shuffling is enough?

A riffle shuffle splits the deck into two halves and interleaves them, as most people do when they play. Mathematicians who have studied the process, notably Persi Diaconis and colleagues, found that around seven riffle shuffles are usually enough to mix a 52-card deck well. Fewer shuffles leave detectable traces of the original order. Casual card players often shuffle only two or three times, which is one reason some hands seem oddly patterned, particularly if the cards were stacked by suit at the end of the last game.

A few practical points:

• Overhand shuffles mix more slowly than riffles and need many more repetitions.
• Casinos use shuffling machines or specific procedures to keep decks well mixed.
• Shuffling does not change the odds of any individual hand. Each card is equally likely to be anywhere in a thoroughly mixed deck.

Why it matters to the mathematics of games

Probability calculations, such as the ones in House Edge and Odds Explained, assume a random, well-mixed deck. If a deck is poorly shuffled, those assumptions weaken. This is one of the reasons why serious games of cards take shuffling seriously.

Games and responsibility

Card games can be a pleasant social pastime. If you play for money, keep in mind that it is for adults only, that laws vary by location and that spending should stay within a budget you set in advance. If gambling stops being fun, take a break and speak to someone you trust or a helpline.

Frequently asked questions

Is it true that every shuffle is unique?

It is extremely likely. With about 8.07 × 10^67 possible orderings, the chance of repeating an order by accident is vanishingly small.

Does shuffling help me predict cards?

No. A properly mixed deck is unpredictable, and each card is equally likely to appear next.

How many shuffles should I do at home?

Seven riffle shuffles is a commonly cited figure for a thorough mix, though a few extra do no harm.